Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| Fitting | C9×#120 ⋊ C2 |
| Sylow | C9 ⋊ H(3,𝔽4) |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C23×D9 |
| commutator, cosocle | C22×C9 | C24 |
| Frattini, socle | C22×C3 | C23×D3 |
| Fitting | C9×#120 | C2 |
| hypercenter | #120 | D9 |
| hypocenter | C9 | H(3,𝔽4) |
| LCS | G ▷ C22×C9 ▷ C9 | |
| UCS | C1 ◁ C22 ◁ #120 | |
390 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 390 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^6,3^2|g^3=h,c^a=cf,d^a=def,g^a=g^2h^2,h^a=h^2,c^b=ce,d^b=df75 conjugacy classes — view the full character table.
| maximal subgroup of | #11798624, #11798633, D9×H(3,𝔽4), … |
| maximal normal subgroup of | #11798624, #11798633, D9×H(3,𝔽4), … |
| maximal quotient of | #11807580, #11807583, #11807589, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 1578);